Partial Homology Relations - Satisfiability in terms of Di-Cographs

Directed cographs (di-cographs) play a crucial role in the reconstruction of evolutionary histories of genes based on homology relations which are binary relations between genes. A variety of methods based on pairwise sequence comparisons can be used to infer such homology relations (e.g.\ orthology, paralogy, xenology). They are \emph{satisfiable} if the relations can be explained by an event-labeled gene tree, i.e., they can simultaneously co-exist in an evolutionary history of the underlying genes. Every gene tree is equivalently interpreted as a so-called cotree that entirely encodes the structure of a di-cograph. Thus, satisfiable homology relations must necessarily form a di-cograph. The inferred homology relations might not cover each pair of genes and thus, provide only partial knowledge on the full set of homology relations. Moreover, for particular pairs of genes, it might be known with a high degree of certainty that they are not orthologs (resp.\ paralogs, xenologs) which yields forbidden pairs of genes. Motivated by this observation, we characterize (partial) satisfiable homology relations with or without forbidden gene pairs, provide a quadratic-time algorithm for their recognition and for the computation of a cotree that explains the given relations

The matroid structure of representative triple sets and triple-closure computation

The closure cl (R) of a consistent set R of triples (rooted binary trees on three leaves) provides essential information about tree-like relations that are shown by any supertree that displays all triples in . In this contribution, we are concerned with representative triple sets, that is, subsets R' of R with cl (R') = cl . In this case, R' still contains all information on the tree structure implied by R, although R' might be significantly smaller. We show that representative triple sets that are minimal w.r.t. inclusion form the basis of a matroid. This in turn implies that minimal representative triple sets also have minimum cardinality. In particular, the matroid structure can be used to show that minimum representative triple sets can be computed in polynomial time with a simple greedy approach. For a given triple set R that “identifies” a tree, we provide an exact value for the cardinality of its minimum representative triple sets. In addition, we utilize the latter results to provide a novel and efficient method to compute the closure cl (R) of a consistent triple set R that improves the time complexity (R Lr 4) of the currently fastest known method proposed by Bryant and Steel (1995). In particular, if a minimum representative triple set for R is given, it can be shown that the time complexity to compute cl (R) can be improved by a factor up to R Lr . As it turns out, collections of quartets (unrooted binary trees on four leaves) do not provide a matroid structure, in general

On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions

Symbolic ultrametrics define edge-colored complete graphs K_n and yield a simple tree representation of K_n. We discuss, under which conditions this idea can be generalized to find a symbolic ultrametric that, in addition, distinguishes between edges and non-edges of arbitrary graphs G=(V,E) and thus, yielding a simple tree representation of G. We prove that such a symbolic ultrametric can only be defined for G if and only if G is a so-called cograph. A cograph is uniquely determined by a so-called cotree. As not all graphs are cographs, we ask, furthermore, what is the minimum number of cotrees needed to represent the topology of G. The latter problem is equivalent to find an optimal cograph edge k-decomposition {E_1,...,E_k} of E so that each subgraph (V,E_i) of G is a cograph. An upper bound for the integer k is derived and it is shown that determining whether a graph has a cograph 2-decomposition, resp., 2-partition is NP-complete

Botswana's Future: Modeling Population and Sustainable Development Challenges in the Era of HIV/AIDS

One element dominates current discussions of Botswana's future, the problem of HIV/AIDS. Currently, Botswana has one of the highest HIV/AIDS prevalence rates in the world. We estimate that during the period from 1993 through 2001 there were around 130,000 deaths from HIV/AIDS. In 2001 alone, we expect that roughly 32,000 Botswana or about 2% of the country's population will die of that disease. Even with significant changes in behavior, our model shows that around 540,000 Botswana, a number equivalent to about one-third of the country's current population, will die of HIV/AIDS during the two-decade period 2002-2021

Namibia's Future: Modeling Population and Sustainable Development Challenges in the Era of HIV/AIDS

Namibia is a young country, having achieved its independence only 11 years ago. It is a land of great potential. The extent to which this potential will be realized is dominated by one issue, the effects of HIV/AIDS. Currently, Namibia has one of the highest HIV/AIDS prevalence rates in the world. In 2001, around 18,000 or about 1% of Namibia's population will die of AIDS. This number will almost double in the next 10 years. Even with significant changes in behavior, our model shows that over half a million Namibians will die of AIDS from 2002 to 2021. This is roughly a third of Namibia's current population. This report summarizes our findings on how HIV/AIDS will influence population, development, and environment interactions in Namibia during the next two decades

Mozambique's Future: Modeling Population and Sustainable Development Challenges

What are the prospects for sustainable development over the next 20 years in Mozambique? Although it looks as if much of the development prospects are determined by such inherently unpredictable events as war, peace, and weather calamities, there are also many changes and patterns which have a long-term stability and which change only slowly over time. For example, socio-demographic changes, such as labor force skills, and population health have a long momentum. These are very important indicators for the economic development potential of a country. Also, although it is impossible to predict a particular year of heavy rains or droughts, there are long time series of weather from which we can calculate the country's vulnerability to single- or multiple-year weather disasters. To focus our efforts in answering this bold question, we concentrate on four issues: (1) Can poverty be erased in the next 20 years? (2) How will school enrollment lead to higher skills in the labor force by 2020? (3) What role will water play in development, in particular, water provision by rain to rural areas, and infrastructure to cities? (4) And, most importantly, what will be the impacts of the HIV/AIDS pandemic in the next decades

Implicit Memory in Music and Language

Research on music and language in recent decades has focused on their overlapping neurophysiological, perceptual, and cognitive underpinnings, ranging from the mechanism for encoding basic auditory cues to the mechanism for detecting violations in phrase structure. These overlaps have most often been identified in musicians with musical knowledge that was acquired explicitly, through formal training. In this paper, we review independent bodies of work in music and language that suggest an important role for implicitly acquired knowledge, implicit memory, and their associated neural structures in the acquisition of linguistic or musical grammar. These findings motivate potential new work that examines music and language comparatively in the context of the implicit memory system

Injective split systems

A split system $\mathcal S$ on a finite set $X$, $|X|\ge3$, is a set of bipartitions or splits of $X$ which contains all splits of the form $\{x,X-\{x\}\}$, $x \in X$. To any such split system $\mathcal S$ we can associate the Buneman graph $\mathcal B(\mathcal S)$ which is essentially a median graph with leaf-set $X$ that displays the splits in $\mathcal S$. In this paper, we consider properties of injective split systems, that is, split systems $\mathcal S$ with the property that $\mathrm{med}_{\mathcal B(\mathcal S)}(Y) \neq \mathrm{med}_{\mathrm B(\mathcal S)}(Y')$ for any 3-subsets $Y,Y'$ in $X$, where $\mathrm {med}_{\mathcal B(\mathcal S)}(Y)$ denotes the median in $\mathcal B(\mathcal S)$ of the three elements in $Y$ considered as leaves in $\mathcal B(\mathcal S)$. In particular, we show that for any set $X$ there always exists an injective split system on $X$, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph $\mathcal B(\mathcal S)$ needs to become in order for a split system $\mathcal S$ on $X$ to be injective. We do this by introducing a quantity for $|X|$ which we call the injective dimension for $|X|$, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map on $X$ can be represented using Buneman graphs.Comment: 22 pages, 3 figure